On a class of partially hypoelliptic microdifferential equations
Let M be a real analytic manifold with a complexification X. We consider the microdifferential equation defined in a neighborhood of \(\rho_ 0\in \dot T^*_ MX\) whose principal symbol is written in the form (1) \(p=p_ 1+\sqrt{-1}q_ 1^{2m}p_ 2\) in a neighborhood of \(\rho_ 0\in \dot T^*_ MX\). Here \(p_ 1\), \(p_ 2\) and \(q_ 1\) are homogeneous holomorphic functions of order 1, 1 and 0 respectively. We assume the following conditions: (2) \(p_ 1\), \(p_ 2\) and \(q_ 1\) are real valued on \(T^*_ MX\), (3) \(dp_ 1\), \(dp_ 2\) and the canonical 1-form \(\omega\) are linearly independent at \(\rho_ 0\), (4) \(\{p_ 1,p_ 2\}=0\) if \(p_ 1=p_ 2=0\), \(\{\), \(\}\) denotes the Poisson bracket on \(T^*_ MX\), (5) \(\{p_ 1,q_ 1\}\neq 0\) at \(\rho_ 0\), (6) \(p_ 1(\rho_ 0)=p_ 2(\rho_ 0)=q_ 1(\rho_ 0)=0.\) In the above situation, we study propagation of singularities of solutions to the equation \(Pu=0\) on the regular involutory submanifold \(\Sigma =\{\rho \in \dot T^*_ MX;\quad p_ j(\rho)=0\quad (1\leq j\leq 2)\}.\) Using a result on second-microhyperbolic operators, we show that the operator P is hypoelliptic second microlocally along \(\Sigma\).
scientific article; zbMATH DE number 4088122
scientific article